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Finkel Problem: Emptiness for one-dimensional PVASS

Establish whether the emptiness problem is decidable for one-dimensional pushdown vector addition systems (PVASS), i.e., valence automata combining a single-dimensional VASS with a pushdown stack. This is the simplest unresolved instance in the SC+ class.

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Background

PVASS extend vector addition systems with a pushdown stack and capture powerful models beyond plain VASS or pushdown automata. The one-dimensional case (single counter plus stack) is a long-standing benchmark known as the Finkel Problem.

The paper highlights that deciding emptiness for SC+ is open and points to one-dimensional PVASS as the simplest open example, underscoring the importance of resolving this specific case to advance the overall classification.

References

In $SC+$, one can arbitrarily alternate between adding $N$-counters and building stacks. Since $Z$-counters can always be simulated by $N$-counters, $SC+$ is more powerful than $SC\pm$, but the decidability of emptiness is open, the simplest open example being one-dimensional PVASSProp.~3.6---also known as the Finkel Problem.

— Verifying Unboundedness via Amalgamation (2405.10296 - Anand et al., 16 May 2024) in Section 1.4, PVASS and their Restrictions